Mathematical Knowledge for Teaching Teachers: The Case of Multiplication and Division of Fractions

This study attempts to answer the question, What is the mathematical knowledge required by teachers of elementary mathematics content courses in the area of multiplication and division of fractions? Beginning in the mid-1980s, when Shulman (1986) introduced the idea of pedagogical content knowledge, researchers have been looking at the knowledge needed to teach in a variety of different content areas. One area that has garnered much of the research is that of mathematics. Researchers have developed frameworks for what they call mathematical knowledge for teaching, but there has been little work done looking at the knowledge requirements for teachers of teachers. This study attempts to fill this gap by determining some aspects of a framework for the mathematical knowledge required to teach prospective elementary teachers multiplication and division of fractions. In order to determine aspects of a framework for mathematics teacher educator knowledge in relation to multiplication and division of fractions, I interviewed, observed, and audiotaped three experienced teacher educators in different educational settings to determine the mathematical work of teaching prospective teachers fraction multiplication and division. My analysis focused on three of major tasks that came out of the work: introducing fraction multiplication, helping students make sense of fraction division, and assessing student understanding. Each of these tasks played a major role in the work of the teacher educators, and the knowledge required to perform these tasks was evident in varying degrees in each teacher educator. After analyzing the three mathematical tasks and the knowledge required by them, I was able to determine some components of a framework for the mathematical knowledge needed for teaching teachers multiplication and division of fractions. These aspects include: understanding multiple representations of fraction multiplication and division and how these representations relate to each other, to whole number ideas, and to the algorithms, deciding which aspects of the topics will help prospective teachers make the connections that they will need in order to teach these topics, especially since time often plays a factor in what gets taught in mathematics content classes for prospective teachers, setting specific goals of exactly what one wants one\u27s students to know, rather than having a general goal of wanting prospective teachers to develop conceptual understanding of a topic, and being able to design and use assessments effectively to help decide if one is achieving one\u27s goals. While each of the aspects described above are components of a framework for the mathematical knowledge needed by teacher educators, the three teacher educators in my study all lacked or were unable to demonstrate some of the knowledge components that would have helped them to meet their goals, despite having a wealth of experience teaching and designing mathematics content courses for prospective elementary teachers. One possible reason for this is that each of the teacher educators in my study were basically alone in their departments, without opportunities to collaborate or discuss these ideas with anyone else. These results suggest a need for better professional development for teacher educators in the field of mathematics education

Using transformative learning theory to help prospective teachers learn mathematics that they already “know”

Prospective Teachers (PTs) often enter their mathematics content courses believing that they know enough mathematics to teach elementary school. However, research has shown that much of PTs’ knowledge is procedurally based and lacks depth and conceptual understanding. One job of Mathematics Teacher Educators (MTEs) in mathematics content courses is to help PTs become more mathematically proficient, by relearning the mathematics that they believe they already know in deeper, more connected ways. We suggest that one way for MTEs to do this is to incorporate Transformative Learning Theory (TLT) into their mathematics content courses. TLT is an application of andragogy, which are the methods or techniques used to teach adults. Through TLT, learners participate in a process where they are presented with a disorienting dilemma that perturbs their prior understandings. Learners work through the dilemma by critically reflecting on their prior understandings and participating in rational discourse with others. Ultimately, learners are tasked with making connections between their prior understandings and their new knowledge. In this paper, we describe a cycle of transformative learning theory and give examples of incorporating TLT into mathematics content courses for PTs through lessons on proportional reasoning and whole number concepts. We conclude by discussing general considerations and resources for incorporating TLT into mathematics content courses and how this helps PTs develop the five strands of mathematical proficiency (National Research Council, 2001)

Prospective Teachers’ Attention to Realism and Consistency with Negative Integers, Addition, and Temperature

Writing and evaluating contextual problems is an important task in the work of teaching, and thus is part of the knowledge that prospective teachers must develop. In dealing with word problems posed both by children and themselves, prospective teachers will need to attend to the realism of the context and the consistency between the operation and context with integer operations. This study describes an examination of the ways in which 100 prospective teachers responded to a child’s temperature story for an integer addition number sentence (i.e., 9 þ 6 ¼ ☐). The child’s story, which was an actual story posed by a Grade 5 student, did not use temperature realistically (realism), nor was it consistent with the given number sentence (consistency). The results indicated that when prospective teachers evaluated the child’s story, they tended to either focus on realism or consistency, but not both. If prospective teachers did not focus their response on realism or consistency, the response was much more likely to be unrealistic or inconsistent itself. Implications point to the importance of addressing both realism and consistency issues when examining integer word problems with prospective teachers

Mathematical Content Knowledge for Teaching Elementary Mathematics: A Focus on Fractions

This article presents a research summary of prospective elementary teachers’ (PTs’) mathematical content knowledge in the area of fractions. The authors conducted an extensive review of the research literature and present the findings across three time frames: a historical look (pre-­‐1998), a current perspective (1998–2011), and a look at the horizon (2011–2013). We discuss 43 articles written across these time frames that focus on PTs’ fraction knowledge. Consistent across these papers is that PTs’ fraction knowledge is relatively strong when it comes to performing procedures, but that they generally lack flexibility in moving away from procedures and using “fraction number sense” and have trouble understanding the meanings behind the procedures or why procedures work. Across the time frames, the trend in the research has moved from looking almost entirely at PTs’ understanding of fraction operations, particularly multiplication and division, to a more balanced study of both their knowledge of operations and fraction concepts. What is lacking in the majority of these studies are ways to help improve upon PTs’ fraction content knowledge. Findings from this summary suggest the need for a broader study of fractions in both content and methods courses for PTs, as well as research into how PTs’ fraction content knowledge develops

Exploring mathematical knowledge for teaching teachers: Supporting prospective elementary teachers’ relearning of mathematics

The growing number of studies on mathematics teacher educator knowledge have consistently argued that mathematics teacher educators require specialized knowledge in their work with prospective teachers (beyond the knowledge needed for teaching students), what researchers refer to as mathematical knowledge for teaching teachers. Drawing from existing research and aspects of our own work as mathematics teacher educators, we offer our own conceptualization of mathematical knowledge for teaching teachers and illustrate ways in which we as mathematics teacher educators use our own knowledge in teaching mathematics content to prospective teachers. We are particularly concerned with the knowledge mathematics teacher educators use to support prospective teachers’ relearning of mathematics, which involves prospective teachers ultimately reconstructing their previously developed knowledge of mathematics. We will illustrate ways in which we use various aspects of mathematical knowledge for teaching teachers to support prospective teachers’ relearning of mathematics through the lens of three different tasks of teaching. We conclude with a discussion of the implications of our analysis for informing the growing knowledge base for mathematics teacher educators

Prospective Elementary Mathematics Teacher Content Knowledge: What Do We Know, What Do We Not Know, and Where Do We Go?

In this Special Issue, the authors reviewed 112 research studies from 1978 to 2012 on prospective elementary teachers’ content knowledge in five content areas: whole numbers and operations, fractions, decimals, geometry and measurement, and algebra. Looking across these studies, this final paper identifies the trends and common themes in terms of the counts and types of studies and commonalities among findings. Analyses of the counts show that the number of articles published each year focusing on prospective teacher (PT) content knowledge is increasing. Most articles across the content areas show that PTs tend to rely on procedures rather than concepts. However, the focus of most articles is identifying PTs’ misconceptions rather than understanding PTs’ conceptions and the development thereof. Both the limitations of the reviews and the directions for future research studies are elaborated

Prospective Elementary Teacher Mathematics Content Knowledge: An Introduction

This Special Issue on the mathematical content knowledge of prospective elementary teachers (PTs) provides summaries of the extant peer-­‐reviewed research literature from 1978 to 2012 on PTs’ content knowledge across several mathematical topics, specifically whole number and operations, fractions, decimals, geometry and measurement, and algebra. Each topic-­‐specific summary of the literature is presented in a self-­‐contained paper, written by a subgroup of a larger Working Group that has collaborated across several years, resulting in this Special Issue sharing the final work. The authors hope this summative look at prospective teacher content knowledge will be of interest to the mathematics education community and will be a useful resource when considering future research as well as designing mathematics content courses for prospective elementary teachers

Prospective Elementary Mathematics Teacher Content Knowledge: What Do We Know, What Do We Not Know, and Where Do We Go?

The authors reviewed 112 research studies from 1978 to 2012 on prospective elementary teachers\u27 content knowledge in five content areas: whole numbers and operations, fractions, decimals, geometry and measurement, and algebra. Looking across these studies, this final paper identifies the trends and common themes in terms of the counts and types of studies and commonalities among findings. Analyses of the counts show that the number of articles published each year focusing on prospective teacher (PT) content knowledge is increasing. Most articles across the content areas show that PTs tend to rely on procedures rather than concepts. However, the focus of most articles is identifying PTs\u27 misconceptions rather than understanding PTs\u27 conceptions and the development thereof. Both the limitations of the reviews and the directions for future research studies are elaborated

Reflective Analysis as a Tool for Task Redesign: The Case of Prospective Elementary Teachers Solving and Posing Fraction Comparison Problems

Mathematical task design has been a central focus of the mathematics education research community over the last few years. In this study, six university teacher educators from six different US institutions formed a community of practice to explore key aspects of task design (planning, implementing, reflecting, and modifying) in the context of comparing fractions using reasoning and sense-making. By presenting results of their implementation of two tasks with 63 prospective elementary teachers across three institutions and their reflective analysis of the implementation, the authors highlight the importance of collecting and analyzing data and reflecting on this analysis to inform the redesign of tasks. The authors also found that considering different types of tasks (problem solving vs. problem posing) helps illuminate different aspects of prospective elementary teachers\u27 understanding, which can inform task redesign. Finally the authors contribute to the knowledge base on reasoning strategies for comparing fractions and prospective elementary teachers’ knowledge of these strategies

Greater Number of Larger Pieces: A Strategy to Promote Prospective Teachers’ Fraction Number Sense Development

Prospective teachers (PTs) need opportunities to develop fraction number sense, yet little research has explicated how this development occurs. Our research team collaboratively designed a task targeted at helping PTs develop fraction number sense through an exploration of fraction comparison strategies. This paper focuses on developing one particular strategy, which we call Greater Number of Larger Pieces (GLP). We argue that understanding this strategy has the potential to support PTs’ number sense, particularly in regards to the measure interpretation of fractions. Analysis of data from two iterations of this task (implemented by five mathematics teacher educators at five US institutions with 124 PTs) showed an improvement in the task’s ability to naturally elicit the GLP strategy from PTs. We share our task, results from each iteration, and discuss modifications that we believe led to increased usage of the GLP strategy